Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.20437 · Physical Review A

Critical spectrum of the anisotropic two-photon quantum Rabi model

Jiong Li, Daniel Braak, Qing‐Hu Chen

Published Apr 4, 2025• 8 citations
arXivHTMLPDF

Abstract

The anisotropic two-photon quantum Rabi model is studied using the Bogoliubov operator approach. Doubly degenerate exceptional states are identified using analytical methods. By adjusting the position of the last exceptional point belonging to two adjacent energy levels, we derive a condition for the absence of the discrete spectrum at the critical coupling where ``spectral collapse'' occurs. In this special case, the spectrum becomes fully continuous above a threshold energy, with no bound states existing, whereas the ground state remains gapped in the general case. More interestingly, we rigorously find that a finite number of bound states exist between two anisotropy-dependent critical atomic frequencies, with infinitely many bound states beyond this frequency regime. In this manner, all issues in the two-photon quantum Rabi model are resolved.

DOI
10.1103/physreva.111.043706

Related papers

Spectral collapse in the two-photon quantum Rabi model

Román J. Armenta‐Rico, F. H. Maldonado-Villamizar, B. M. Rodríguez-Lara

Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.

arXiv:1912.09030•Physical Review A•Jun 18, 2020•44 citations

Spectral collapse in anisotropic two-photon Rabi model

Chi‐Fai Lo

In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse in the anisotropic two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping (together with the anisotropy) effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it. The light-matter interaction appears to dominate the frequency modulation as its coupling strengths go beyond the critical values, leading to the emergence of the spectral collapse. However, at the critical couplings the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. Accordingly, at the critical couplings the eigenenergy spectrum comprises both a set of discrete energy levels and a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in a finite potential well, and the number of bound states available is determined by the energy difference between the two atomic levels. Each of these eigenenergies has a twofold degeneracy corresponding to the spin degree of freedom.

Scientific Reports•Jun 11, 2021•3 citations

Analytical solution and spectral structure of the two-photon anisotropic Rabi-Stark model

Zhan-Yuan Yan, Jiaxin Cheng, Fangcheng Qiu +3 more

Abstract Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.

Physica Scripta•May 28, 2024•3 citations

Critical Spectrum and Quantum Criticality in the Two‐Photon Rabi–Stark Model

Jiong Li, Qing‐Hu Chen

ABSTRACT We investigate the spectral properties and quantum criticality of the two‐photon Rabi–Stark model. Using the exact solution of this model, we rigorously derive a condition for complete spectral collapse, where all bound states vanish. In this case, the energy gap closes at a critical coupling, signaling a continuous quantum phase transition. The corresponding gap exponent differs from those in both the one‐photon Rabi–Stark model and the quantum Rabi model, suggesting a distinct universality class. While in the general case, an infinite number of discrete bound states exist when spectral collapse occurs and the energy gap remains open. By mapping to an inverse square potential well, these bound levels approach the threshold energy exponentially. Our results offer new insights into novel spectral phenomena in nonlinear quantum Rabi models, with potential implications for experimental realizations in circuit quantum electrodynamics and trapped ion systems.

arXiv:2505.20703•Advanced Quantum Technologies•Nov 11, 2025•2 citations

Demystifying the spectral collapse in two-photon Rabi model

Chi‐Fai Lo

We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature "spectral collapse", by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a "Lorentzian function" potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the "spectral collapse" can be monitored in a straightforward manner.

Scientific Reports•Sep 9, 2020•14 citations

Quantum Criticality from Spectral Collapse in the Two-Photon Rabi Model

Jiong Li, Jun-Ling Wang, Qing-Hu Chen +1 more

Spectral collapse in the two-photon quantum Rabi model (tpQRM) has long been regarded as insufficient to establish quantum criticality because, under generic conditions, the lowest excitation gap remains finite. We show that, at a special qubit frequency, spectral collapse in the anisotropic tpQRM constitutes a continuous quantum phase transition governed by a single soft mode. The same-parity gap, set by the vanishing effective oscillator frequency, closes with exponent $zν= 1/2$, whereas the lower different-parity gap arises from symmetry-induced splitting. Parity symmetry excludes the different-parity excitation from the quantum Fisher information response and restricts Kibble-Zurek dynamics to excitations within the ground-state parity sector. Together, these results identify the same-parity excitation as the soft mode and its gap as the characteristic energy scale. The corresponding critical exponents, $ν=1/4$ and $z=2$, place the anisotropic tpQRM in the same universality class as the standard QRM. Our results establish spectral collapse as a potential route to experimentally accessible quantum criticality in a few-body system and show that universality is determined by the soft-mode structure rather than by the closing of the lowest excitation gap.

quant-pharXiv:2604.23164•Physical Review Letters•Apr 28, 2026•0 citations
The anisotropic two-photon quantum Rabi model is studied using the Bogoliubov operator approach. Doubly degenerate exceptional states are identified using analytical methods. By adjusting the position of the last exceptional point belonging to two adjacent energy levels, we derive a condition for the absence of the discrete spectrum at the critical coupling where ``spectral collapse'' occurs. In this special case, the spectrum becomes fully continuous above a threshold energy, with no bound states existing, whereas the ground state remains gapped in the general case. More interestingly, we rigorously find that a finite number of bound states exist between two anisotropy-dependent critical atomic frequencies, with infinitely many bound states beyond this frequency regime. In this manner, all issues in the two-photon quantum Rabi model are resolved.
Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.
In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse in the anisotropic two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping (together with the anisotropy) effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it. The light-matter interaction appears to dominate the frequency modulation as its coupling strengths go beyond the critical values, leading to the emergence of the spectral collapse. However, at the critical couplings the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. Accordingly, at the critical couplings the eigenenergy spectrum comprises both a set of discrete energy levels and a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in a finite potential well, and the number of bound states available is determined by the energy difference between the two atomic levels. Each of these eigenenergies has a twofold degeneracy corresponding to the spin degree of freedom.
Abstract Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.
ABSTRACT We investigate the spectral properties and quantum criticality of the two‐photon Rabi–Stark model. Using the exact solution of this model, we rigorously derive a condition for complete spectral collapse, where all bound states vanish. In this case, the energy gap closes at a critical coupling, signaling a continuous quantum phase transition. The corresponding gap exponent differs from those in both the one‐photon Rabi–Stark model and the quantum Rabi model, suggesting a distinct universality class. While in the general case, an infinite number of discrete bound states exist when spectral collapse occurs and the energy gap remains open. By mapping to an inverse square potential well, these bound levels approach the threshold energy exponentially. Our results offer new insights into novel spectral phenomena in nonlinear quantum Rabi models, with potential implications for experimental realizations in circuit quantum electrodynamics and trapped ion systems.
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature "spectral collapse", by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a "Lorentzian function" potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the "spectral collapse" can be monitored in a straightforward manner.
Spectral collapse in the two-photon quantum Rabi model (tpQRM) has long been regarded as insufficient to establish quantum criticality because, under generic conditions, the lowest excitation gap remains finite. We show that, at a special qubit frequency, spectral collapse in the anisotropic tpQRM constitutes a continuous quantum phase transition governed by a single soft mode. The same-parity gap, set by the vanishing effective oscillator frequency, closes with exponent zν=1/2zν= 1/2zν=1/2, whereas the lower different-parity gap arises from symmetry-induced splitting. Parity symmetry excludes the different-parity excitation from the quantum Fisher information response and restricts Kibble-Zurek dynamics to excitations within the ground-state parity sector. Together, these results identify the same-parity excitation as the soft mode and its gap as the characteristic energy scale. The corresponding critical exponents, ν=1/4ν=1/4ν=1/4 and z=2z=2z=2, place the anisotropic tpQRM in the same universality class as the standard QRM. Our results establish spectral collapse as a potential route to experimentally accessible quantum criticality in a few-body system and show that universality is determined by the soft-mode structure rather than by the closing of the lowest excitation gap.